Abstract: By examining the linkage between conservation laws and symmetry, we explain why
it appears there should not be an analogue of a complete integral for the
Hamilton-Jacobi equation for integrable nonholonomic systems.

Abstract: In this paper, we describe a geometric setting for higher-order La- grangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.

Abstract: Related to the components of the quaternionic Hopf mapping, we propose a parametric Hamiltonian function in $\mathbb{T}^*\mathbb{R}^4$ which is a homogeneous quartic polynomial with six parameters, defining an integrable family of Hamiltonian systems. The key feature of the model is its nested Hamiltonian-Poisson structure, which appears as two extended Euler systems in the reduced equations. This is fully exploited in the process of integration, where we find two 1-DOF subsystems and a quadrature involving both of them. The solution is quasi-periodic, expressed by means of Jacobi elliptic functions and integrals, based on two periods. For a suitable choice of the parameters, some remarkable classical models such as the Kepler, geodesic flow, isotropic oscillator and free rigid body systems appear as particular cases.

Abstract: The Lagrangian description of mechanical systems and the Legendre Transformation
(considered as a passage from the Lagrangian to the Hamiltonian
formulation of the dynamics) for point-like objects, for which the
infinitesimal configuration space is $T M$, is based on the existence of
canonical symplectic isomorphisms of double vector bundles $T^* TM$, $T^*T^* M$,
and $TT^* M$, where the symplectic structure on $TT^* M$ is the tangent lift of the canonical symplectic structure $T^* M$.
We show that there exists an analogous picture in the dynamics of objects for which the configuration space is $\wedge^n T M$, if we make use of certain structures of graded bundles of degree $n$, i.e. objects generalizing vector bundles (for which $n=1$). For instance, the role of $TT^*M$ is played in our approach by the manifold $\wedge^nT M\wedge^nT^*M$, which is canonically a graded bundle of degree $n$ over $\wedge^nT M$. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.

Abstract: In this paper we have obtained some dynamics equations, in the presence of
nonlinear nonholonomic constraints and according to a lagrangian and some
Chetaev-like conditions. Using some natural regular conditions, a simple
form of these equations is given. In the particular cases of linear and
affine constraints, one recovers the classical equations in the forms known
previously, for example, by Bloch and all [3,4]. The case of time-dependent
constraints is also considered. Examples of linear constraints, time
independent and time depenndent nonlinear constraints are considered, as
well as their dynamics given by suitable lagrangians. All examples are
based on classical ones, such as those given by Appell's machine.

Abstract: We prove the differentiability of Mather's average action on all
rotation vectors of measures whose supports are contained in a
Lipschitz Lagrangian asymptotically isolated graph, invariant by
Tonelli Hamiltonians. We also show the relationship between
differentiability of $\beta $ and local integrability of the
Hamiltonian flow.